Example 44.11 Exact Poisson Regression

The following data, taken from Cox and Snell (1989, pp. 10–11), consists of the number, Notready, of ingots that are not ready for rolling, out of Total tested, for several combinations of heating time and soaking time:

The following invocation of PROC GENMOD fits an asymptotic (unconditional) Poisson regression model to the data. The variable
Notready is specified as the response variable, and the continuous predictors Heat and Soak are defined in the CLASS statement as categorical predictors that use reference coding. Specifying the offset variable as
lnTotal enables you to model the ratio Notready/Total.

The EXACT statement is specified to additionally fit an exact conditional Poisson regression model. Specifying the lnTotal offset variable models the ratio Notready/Total; in this case, the Total variable contains the largest possible response value for each observation. The JOINT
option produces a joint test for the significance of the covariates, along with the usual marginal tests. The ESTIMATE
option produces exact parameter estimates for the covariates. The STATUSTIME=10
option is specified in the EXACTOPTIONS
statement for monitoring the progress of the results; this example can take several minutes to complete due to the JOINT
option. If you run out of memory, see the SAS Companion for your system for information about how to increase the available
memory.

The "Criteria For Assessing Goodness Of Fit" table is displayed in Output 44.11.1. Comparing the deviance of 10.9363 to an asymptotic chi-square distribution with 11 degrees of freedom, you find that the
p-value is 0.449. This indicates that the specified model fits the data reasonably well.

Output 44.11.1: Unconditional Goodness of Fit Criteria

The GENMOD Procedure

Criteria For Assessing Goodness Of Fit

Criterion

DF

Value

Value/DF

Deviance

11

10.9363

0.9942

Scaled Deviance

11

10.9363

0.9942

Pearson Chi-Square

11

9.3722

0.8520

Scaled Pearson X2

11

9.3722

0.8520

Log Likelihood

-7.2408

Full Log Likelihood

-12.9038

AIC (smaller is better)

41.8076

AICC (smaller is better)

56.2076

BIC (smaller is better)

49.3631

From the "Analysis Of Parameter Estimates" table in Output 44.11.2, you can see that only two of the Heat parameters are deemed significant. Looking at the standard errors, you can see that
the unconditional analysis had convergence difficulties with the Heat=7 parameter (Standard Error=264324.6), which means you
cannot fit this unconditional Poisson regression model to this data.

Output 44.11.2: Unconditional Maximum Likelihood Parameter Estimates

Analysis Of Maximum Likelihood Parameter Estimates

Parameter

DF

Estimate

StandardError

Wald 95% Confidence Limits

Wald Chi-Square

Pr > ChiSq

Intercept

1

-1.5700

1.1657

-3.8548

0.7147

1.81

0.1780

Heat

7

1

-27.6129

264324.6

-518094

518039.0

0.00

0.9999

Heat

14

1

-3.0107

1.0025

-4.9756

-1.0458

9.02

0.0027

Heat

27

1

-1.7180

0.7691

-3.2253

-0.2106

4.99

0.0255

Soak

1

1

-0.2454

1.1455

-2.4906

1.9998

0.05

0.8304

Soak

1.7

1

0.5572

1.1217

-1.6412

2.7557

0.25

0.6193

Soak

2.2

1

0.4079

1.2260

-1.9951

2.8109

0.11

0.7394

Soak

2.8

1

-0.1301

1.4234

-2.9199

2.6597

0.01

0.9272

Scale

0

1.0000

0.0000

1.0000

1.0000

Note:

The scale parameter was held fixed.

Following the output from the asymptotic analysis, the exact conditional Poisson regression results are displayed, as shown
in Output 44.11.3.

Output 44.11.3: Exact Tests

The GENMOD Procedure

Exact Conditional Analysis

Exact Conditional Tests

Effect

Test

Statistic

p-Value

Exact

Mid

Joint

Score

18.3665

0.0137

0.0137

Probability

1.294E-6

0.0471

0.0471

Heat

Score

15.8259

0.0023

0.0022

Probability

0.000175

0.0063

0.0062

Soak

Score

1.4612

0.8683

0.8646

Probability

0.00735

0.8176

0.8139

The Joint test in the "Conditional Exact Tests" table in Output 44.11.3 is produced by specifying the JOINT
option in the EXACT
statement. The p-values for this test indicate that the parameters for Heat and Soak are jointly significant as explanatory effects in the model. If the Heat variable is the only explanatory variable in your model, then the rows of this table labeled as "Heat" show the joint significance
of all the Heat effect parameters in that reduced model. In this case, a model that contains only the Heat parameters still explains a significant amount of the variability; however, you can see that a model that contains only the
Soak parameters would not be significant.

The "Exact Parameter Estimates" table in Output 44.11.4 displays parameter estimates and tests of significance for the levels of the CLASS variables. Again, the Heat=7 parameter
has some difficulties; however, in the exact analysis, a median unbiased estimate is computed for the parameter instead of a maximum likelihood estimate. The confidence limits show that the Heat variable contains some explanatory power, while the categorical Soak variable is insignificant and can be dropped from the model.

Output 44.11.4: Exact Parameter Estimates

Exact Parameter Estimates

Parameter

Estimate

StandardError

95% Confidence Limits

Two-sided p-Value

Heat

7

-2.7552

*

.

-Infinity

-0.7864

0.0199

Heat

14

-3.0255

1.0128

-5.7450

-0.6194

0.0113

Heat

27

-1.7846

0.8065

-3.6779

0.2260

0.0844

Soak

1

-0.3231

1.1717

-2.8673

3.6754

1.0000

Soak

1.7

0.5375

1.1284

-1.8056

4.4588

1.0000

Soak

2.2

0.4035

1.2347

-2.5785

4.5054

1.0000

Soak

2.8

-0.1661

1.4214

-4.5490

4.2168

1.0000

Note:

* indicates a median unbiased estimate.

Note: If you want to make predictions from the exact results, you can obtain an estimate for the intercept parameter by specifying
the INTERCEPT keyword in the EXACT
statement. You should also remove the JOINT
option to reduce the amount of time and memory consumed.